I always thought of the term as nothing more than a poor translation from French. The adjective particulière connotes strangeness without the bluntness of the English word "strange" and the literal cognate of "particular" doesn't insinuate quite enough. Perhaps it could have been better translated as Peculiar Friendships or Certain Friendships, but even those lacks a certain warmth. From Wikipedia we get a note saying...The phrase "les Amitiés particulieres" appears to have been coined in 1724 by the Jesuit priest Father Joseph-François Lafitau (1681-1746) in his study Moeurs des sauvages ameriquains, comparées aux moeurs des premier temps (Paris 1724, vol. 1, pp. 603-09): "L’Athenrosera, ou les Amitiés particulières [sic] entre les jeunes gens..." In the translation by W.N. Fenton & E.L. Moore as Customs of the American Indian compared with the customs of primitive times. (See vol. 1, Toronto, 1974, pp. 361, 364.) "The Athenrosera, or particular friendships, between young people which prevail almost in the same way from one end of America to the other...Among the North American Indians these relationships of friendship carry no suspicion of apparent vice, although there is or may be much real vice. They are very ancient in their origin, very clear in their constant usage, sacred, if I dare say so, in the union which they form, the knots of which are as closely tied as those of blood and nature and can be broken only if one of them, making himself unworthy by cowardly acts which would dishonour his friend, [should] force him to renounce his alliance...The parents are the first to encourage these friendships and to respect their rights...the two become companions in hunting, warfare, and good or bad fortune; they are entitled to food and shelter in each other's lodging."
And Rictor Norton's glossary of 18th century terms is largely composed of biblicisms and vulgarities with a few more colorful terms thrown in, but nothing as subtle as "special friendship."

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